Shadow Catcher wrote:Orange cleaners (Goo Gone) often work well.
Shadow Catcher is correct.

I use Goo Gone on all kinds of price tag glue and other sticker glue, on all kinds of materials without problems. Read and follow instructions and test inconspicuously, before using somewhere that will show, just in case, though. Do not use on silk, satin, wool, leather, suede, or vinyl. If any stubborn spots are left, follow with dish soap and a moist sponge, the kind of two sided kitchen or bath sponge that is reported to not scratch. Wipe dry with a soft cloth or towel. As far as the contact cement goes, you know what you used for that and I would call that manufacturer and ask them.
Also, being in the medical profession, you probably know that baby oil will help remove some sticker glues, like medical tapes. If you are nervous about it, you could try that first and see if it works for this but you will need to wash and rinse it well before putting new weather stripping on.
Many people leave the old stubborn sticky weather stripping residue there and just put the new over it. You can rub and roll a lot of it off, so that there aren't any bumps built up.
Do NOT use Goof Off to remove sticker glue. It WILL remove the outer layers of your finish, especially paints and some other types of finishes.
If you have the old fake marble sink counters and they have any of those cigarette "toast" burns on them, Goof Off will remove the discoloring and smooth over the burns to blend it all in like it never happened. (A tip from my fixer-upper days.)
Regardless of what you use to get the old residue and glue off, wash and dry it well afterwards. Then, follow up by cleaning the area with alcohol or window cleaner and dry with a clean cloth. I have found that cleaning an area with either window cleaner (eg. Windex) or alcohol often removes properties that keep weather stripping from sticking. However, in areas that rub, all bets are off on keeping the weather stripping stuck, unless you tap in a hidden brad or screw with a washer near that spot.
Please let us know your outcome. Thanks.